TREASURIES-Bargain-hunting boosts prices before 3-yr auction

* Buyers find value after sharp post-employment sell-off

* Yield curve steepens after two-day flattening

* Treasury to sell $35 billion in 3-year notes

* Ten-, 30-yr Treasury auctions due Wednesday, Thursday (Updates comments, prices)

By Ellen Freilich

NEW YORK, June 9 (Reuters) - U.S. Treasuries prices rose on Tuesday as buyers found value after the sell-off that followed Friday's news that U.S. employers had cut far fewer jobs in May than expected.

The yield curve, after a dramatic flattening in the two prior sessions based on the idea that the Federal Reserve could raise interest rates next year if the recession ends, steepened again on Tuesday after analysts said the economy was not strong enough for the Fed to raise interest rates.

"The Treasury market has moved to short-term attractive levels; no one thinks the Fed will tighten (monetary policy) any time soon," said Andrew Brenner, senior vice president at MF Global in New York.

Two-year notes US2YT=RR, hit in recent days by fears the Fed could raise interest rates next year, rose 6/32, their yields tumbling to 1.31 percent from 1.41 percent on Monday.

"There's a fairly smooth steepening across the yield curve, but the move pales in comparison to the dramatic flattening of the prior two trading sessions," said John Canavan, analyst at Stone & McCarthy Research Associates in Princeton, New Jersey.

The two days of price-cutting drew buyers despite imminent supply.

The Treasury will auction $35 billion in three-year notes at 1 p.m. EDT (1700 GMT) and 10- and 30-year Treasuries on Wednesday and Thursday, respectively.

"Today's auction should have a reasonable underwriting given the damage that has been done to the front end," said John Spinello, chief fixed-income technical analyst at Jefferies & Co in New York.

"Foreign participation should provide a solid indirect bid," he added.

The recent cheapening of the three-year notes relative to other maturities should allow a reasonable arbitrage bid as well, Spinello said.

The three-year notes to be sold at 1 p.m. EDT (1700 GMT), yielded 1.988 percent in when-issued trade.

The "violent unwinding" of curve trades since Friday's employment report could also enhance the attractiveness of the three-year notes to be sold on Tuesday relative to the 10- and 30-year securities set for sale later in the week, he said.

Analysts said selling in the 10-year note was likely to emerge when its yield fell into the 3.79 percent to 3.77 percent area, while buying would emerge when the yield was in the 3.88 percent to 3.925 percent area.

Spinello said the 10-year yield could reach the 4 percent to 4.07 percent range at the end of this week's auctions, weakness that would present a buying opportunity.

Benchmark 10-year notes US10YT=RR rose 5/32 in price, their yields easing to 3.85 percent from 3.87 percent late on Monday. Thirty-year bonds US30YT=RR were down 10/32, their yields rising to 4.635 percent from 4.61 percent on Monday. (Editing by Dan Grebler)